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Physics LibreTexts

2.5: Impedance

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We need to remind ourselves of one other thing from electromagnetic theory before we can proceed, namely the meaning of impedance in the context of electromagnetic wave propagation. The impedance Z is merely the ratio E/H of the electric to the magnetic field. The SI units of E and H are V/m and A/m respectively, so the SI units of Z are V/A, or ohms, Ω. We are now going to see if we can express the impedance in terms of the permittivity and permeability of the medium in which an electromagnetic wave is travelling.

Maxwell’s equations are

D=ρB=0.×H=˙D+J.×E=˙B.

In an isotropic, homogeneous, nonconducting, uncharged medium (such as glass, for example), the equations become:

E=0H=0×H=ϵ˙E.×H=μ˙H.

If you eliminate H from these equations, you get

2E=ϵμ¨E,

which describes an electric wave of speed

1ϵμ.

In free space, this becomes

1ϵ0μ0

which is 2.998×108ms1.

The ratio of the speeds in two media is

v1v2=n2n1=ϵ2μ2ϵ1μ1,

and if, as is often the case, the two permeabilities are equal (to μ0), then

v1v2=n2n1=ϵ2ϵ1.

In particular, if you compare one medium with a vacuum, you get: n=ϵϵ0.

Light is a high-frequency electromagnetic wave. When a dielectric medium is subject to a high frequency field, the polarization (and hence D) cannot keep up with the electric field E. D lags behind E. This can be described mathematically by ascribing a complex value to the permittivity. The amount of lag depends, unsurprisingly, on the frequency - i.e. on the color - and so the permittivity and hence the refractive index depends on the wavelength of the light. This is dispersion.

If instead you eliminate E from Maxwell’s equations, you get

2H=ϵμ¨H.

This is a magnetic wave of the same speed.

If you eliminate the time between ??? and ???, you find that EH=μϵ, which, in free space, has the value μ0ϵ0=377Ω, which is the impedance of free space. In an appropriate context I may use the symbol Z0 to denote the impedance of free space, and the symbol Z to denote the impedance of some other medium.

The ratio of the impedances in two media is

Z1Z2=ϵ2μ1ϵ1μ2,

and if, as is often the case, the two permeabilities are equal (to μ0), then

Z1Z2=ϵ2ϵ1=n2n1=v1v2.

We shall be using this result in what follows.


This page titled 2.5: Impedance is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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